Congrats! Welcome to Planescape!
Damn, deleted a portion of this post because what started as a recommendation for your next PS purchase got bloated real quick. But, to harp on them anyway:
You don't NEED these at all, but if you do get the urge to go hunting...
Planescape Monstrous Compendium - If you're looking for PS flavor for your creatures. Plus, the absolute best take on the fiends ever, IMO.
Into The Cage - An in-depth examination of Sigil.
Uncaged: Faces of Sigil - This book is nothing but NPCs and interweaving plotlines to introduce to your PCs. One of Tony DiTerlizzi's most impressive bodies of work as well!
Planewalker's Handbook - Great book to hand over to your players to really get 'em into the setting.
Factol's Manifesto - In-depth look at the Factions, their beliefs, organization, goals, schisms, etc.
The Planes boxed sets(Planes of Law, Planes of Conflict, Planes of Chaos) are great as well, but I'd only push them if you really wanted to bounce your players around the planes a LOT and wanted to do it as-written. Lots of great stuff in them, but it isn't critical to have. They're also likely some of the pricier items to track down....
Unfortunately, since you
have to hunt down hardcopies now, some of these are rather unfairly priced.
Planewalker.com is free though!